Pore size distributions of complex systems - Annales UMCS

Pore size distributions of complex systems - Annales UMCS

ANNALES
UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA
LUBLIN – POLONIA
VOL. LX, 16
SECTIO AA
2005
Pore size distributions of complex systems
1,*
2
V.M. Gun’ko and R. Leboda
Institute of Surface Chemistry, 17 General Naumov Street,
03164 Kiev, Ukraine
2
Maria Curie-Skłodowska University, 20-031 Lublin, Poland
1
An approach based on the sum of integral adsorption isotherm equations
including independent distribution functions for each component of
complex adsorbents and solved by using self-consistent regularization was
developed and tested for mechanical mixtures of carbon and silica
adsorbents. For individual adsorbents, e.g. carbons or silica gels, the
developed procedure gives the pore size distributions close to those
calculated using the DT method.
1. INTRODUCTION
Determination of structural and adsorption characteristics of any adsorbent is
a non-trivial problem [1-4], not to mention complex or hybrid adsorbents
because the topology of pores of different components of, e.g., carbon-mineral
adsorbents or composites with polymers and metal oxides is strongly different,
as well as the surface potentials [5-8]. The use of the same adsorption potential
and the same pore model for different components of hybrid adsorbents leads
clearly to significant errors in the determined parameters. Previously we used the
sum of integral equations solved to obtain the pore size distributions for complex
adsorbents using the sum of the integral adsorption equations with one
distribution function [5-7]. The aim of this paper is to show an improved
pathway to describe complex adsorbents using independent distribution
functions of the pore size for each phase and the solution of the sum of integral
adsorption equations using self-consisting regularization with the control of the
correspondence of the pore model to the real pore topology using a criterion
linked to deformation of the pore shape compared with the model [6].
Pore size distributions of complex systems
247
2. COMPUTATIONAL PROCEDURE
The pore size distributions (PSDs) fV(x) (differential distribution function
fV(x) ∼ dVp/dx) of carbon adsorbents can be calculated with overall equation in
the form proposed by Nguyen and Do (ND method) [9, 10]:
a=
rk ( p )
fV ( x)dx +
rmin
rmax
rk ( p )
w
t ( p, x) fV ( x)dx
x − σ sf / 2
(1)
where rmin and rmax are the minimal and maximal half-widths of pores
respectively; w = 1 for slitlike pores; rk(p) is determined by the modified Kelvin
equation:
rk ( p ) =
σ sf
2
+ t ( p, x ) +
wγν m cosθ
,
RgT ln( p0 / p )
(2)
and t(p,x) can be computed using the modified BET equation:
t ( p, x ) = t m
cz [1 + (nb / 2 − n / 2) z n−1 − (nb + 1) z n + (nb / 2 + n / 2) z n+1 ] ,
(1 − z ) [1 + (c − 1) z + (cb / 2 − c / 2) z n − (cb / 2 + c / 2) z n+1 ]
(3)
where
tm = am/SBET,
(4)
b = exp(∆ε/RgT),
(5)
c = cs exp((Q p − Qs ) / RgT ) ,
(6)
cS = γe
E −QL
Rg T
,
(7)
∆ε is the excess of the evaporation heat due to the interference of the layering on
the opposite pore wall (∆ε ≈ 2.2 kJ/mol [9, 11]); t(p,x) is the statistical thickness
of the adsorbed layer; am is the BET monolayer capacity; cs is the BET
adsorption coefficient on flat surface, QL is the liquefaction heat, E is the
adsorption energy, γ is a constant; Qs and Qp are the adsorption heat on flat
248
V. M. Gun’ko and R. Leboda
surface and in pores respectively; z = p/p0; n is the number (non-integer) of
statistical monolayers of adsorbate molecules, and its maximal value for a given
pore half-width x is equal to (x − σsf/2)/tm; and σsf = (σs + σf)/2 is the average
collision diameter of surface (carbon) and fluid (nitrogen) atoms. These
equations can be modified to be used for adsorbents characterized by, for
instance, cylindrical pores (silica gels) or gaps between spherical particles
(fumed oxides, carbon black). Different surface potentials were used on
calculations of Qs and Qp for nitrogen in slitlike pores (Steele potential) [9, 10],
gaps between spherical particles [6, 7], and cylindrical pores (Lennard–Jones
potential) [12-15]. Steele potential was used for the calculations of Qs and Qp for
nitrogen molecule in slitlike pores [9, 10]
U ( x, y ) = ϕ ( y ) + ϕ ( x − y ) ,
(8)
with
ϕ ( y ) = 4πρ sσ ε sf ∆[0.2(
2
sf
σ sf
y
) − 0.5(
10
σ sf
y
) −
4
σ sf4
6∆( y + 0.61∆)3
],
(9)
∆ = 0.3354 nm is the thickness of a nitrogen monolayer, and y is the distance
from the central plane of the outermost atom layer of one pore wall. The solidfluid interaction in cylindrical pores can be determined by [12]
U (r , R) = π 2 ρ sε sf σ sf2 [
r
r
63 r
[
(2 − )]−10 F [−4.5,−4.5,1, (1 − ) 2 ]
R
R
32 σ sf
r
r
r
− [3[
(2 − )]− 4 F [−1.5,−1.5,1, (1 − ) 2 ]
R
R
σ sf
(10)
where F[α,β,γ,χ] is the hypergeometric series, r is the radial coordinate, εsf is the
surface-fluid parameter in the LD potential, and ρs is the density of surface
atoms. In the case of pores as gaps between spherical particles, eq 2 should be
used in the form
ln
p0 γvm 1
2
=
−
,
2
p RgT rk
(R + t'
+ rk ) − R 2 − rk + R + t '
where R is the radius of nanoparticles, and t’ = t + σsf/2.
(11)
Pore size distributions of complex systems
249
The nitrogen desorption or adsorption data can be utilized to compute fV(x)
distributions with eq 1 using regularization procedure [16] CONTIN [17, 18]
modified to the mentioned equations (i.e. modified ND-CONTIN (MNDC)
method) under non-negativity condition (fV(x) ≥ 0 at any x) with a fixed or nonfixed regularization parameter α.
To consider different types of porosity (slitlike and cylindrical pores or gaps
between spherical nanoparticles) simultaneously, integral equation 1 can be rewritten as follows
aΣ =
rk ,i ( p )
ci ai =
i
fV ,i ( x)dx +
ci
i
rmax,i
rmin
w
ti ( p, x) fV , i ( x)dx ,
−
/
2
σ
x
sf
rk ,i ( p )
(12)
where ci = cslit, ccyl and csph are weight constants determining contributions of
slitlike and cylindrical pores or gaps between spherical particles to the total
adsorption (i.e. porosity), using the corresponding modified Kelvin equations
(eqs 2 or 11). Eq 12 could be solved using two approaches: (i) fV,slit(x) = fV,sph(x)
= fV,cyl(x) = fV(x) (i.e. monoregularization with respect to overall fV(x) using the
MNDC method); and (ii) fV,slit(x) ≠ fV,sph(x) ≠ fV,cyl(x) with binary or ternary selfconsistent (subsequent for fV,t(x) at i = slit, cyl, and sph) regularization with
respect to different types of pores (initial fV(x) could be calculated with the
monoregularization) [5, 16].
The fV(x) distributions determined with eqs 1 or 12 and linked to the pore
volume can be transformed to the distributions fS(x) with respect to the specific
surface area using the corresponding models of pores
f S ( x) =
V
w
( fV ( x) − p ) ,
x
x
(13)
where w = 1, 2, and 3 for slitlike, cylindrical, and spherical pores respectively.
However, the relationship for fS(x) and fV(x) is more complex for pores as gaps
between spherical particles, since the inner volume of aggregates of primary
particle plays the role of pores but both outer and inner surfaces of these
aggregates contribute the specific surface area. For a cubic lattice with spherical
nanoparticles w ≈ 1.36; however, this value increases for a denser hexagonal
lattice. For estimation of deviation of the pore shape from slitlike one ∆wslit =
SBET/Ssum,slit – 1, eq 13 could be used at w = 1 for calculations of fS(x) for the
model of slitlike pores with [6]
S sum, slit =
x max
x min
f S ( x)dx =
x max
x min
V
w
( fV ( x) − p )dx ,
x
x
(14)
250
V. M. Gun’ko and R. Leboda
In the case of the mixture of pores
∆wtotal =
Rmax
i
Rmin
S BET
V
wi
( fV ,i ( R) − p )dR
R
R
− 1,
(15)
Different versions of the described approach were used for investigations of
various individual (carbons, silica gels, fumed silicas) and complex adsorbents
such as carbon-mineral and polymer-mineral composites [5-8, 18-37]. In this
paper we used two complex systems with mechanically mixed (I) fumed silica
A-300 (Pilot plant of the Institute of Surface Chemistry, Kalush, Ukraine; SBET =
232 m2/g, Vp = 0.557 cm3/g) and activated carbon PS1 (PSO MASKPOL, Poland,
SBET = 877 m2/g, Vp = 0.445 cm3/g) as 1 : 1 (Fig. 1); and (II) fumed silica A-300,
silica gel Si-60 (Merck, SBET = 447 m2/g, Vp = 0.8 cm3/g), and graphitized carbon
black Envicarb (Supelco, USA, SBET = 98 m2/g, Vp = 0.447 cm3/g) as 1 : 1: 1
(Fig. 2).
Individual carbon adsorbents WVA (wood based activated carbon, Westvaco)
[19, 30] and Carboxen 569 (carbon sieve, Supelco) [6] were used to compare
results of calculations using DFT [38] and MNDC methods.
3. RESULTS AND DISCUSSION
Pore size distributions (PSD) were calculated using the sum of integral
equations (12) corresponding to each component using a complex model of
slitshaped (labeled Slit) pores for activated carbon and carbon black; cylindrical
pores (labeled Cyl) for silica gel, and gaps between spherical particles (labeled
Sph) for fumed silica. Additionally, for the mixture I, self-consistent
regularization (SCR) was used. The comparison of the PSDs of individual
adsorbents and their mixtures (Fig. 2) shows that application of a simple model
of pores can give inappropriate distribution functions; for instance, the model of
slitshaped pores gives the PSD whose peaks are displaced in comparison with
the corresponding peaks of individual adsorbents (we assume that the
individuality of these adsorbents remains because their mechanical mixing was
careful). On the other hand, application of the complex model including pore
models corresponding to all the components of the mixture gives the PSD while
maintaining shapes of the corresponding components (Figs 1 and 2).
However, the PSD of the mixture and the corresponding PSD of individual
components are not identical because mixing and pre-treating (degassing at
200oC) samples can slightly change their complex texture and morphology.
Pore size distributions of complex systems
A-300/PS1
Sph/Slit
Sph/Slit(SCR)
Individual
PS1(Slit)
A-300(Sph)
0.02
IPSDV (a.u.)
251
0.01
0.00
0.2
1
10
100
Pore Radius (nm)
Fig. 1. Incremental PSDs for mechanical mixture of fumed silica A-300 and activated
carbon PS1 (1 : 1) calculated using a complex model of pores with the contribution of
gaps between spherical particles (A-300) and slitshaped pores (PS1) with SCR or
standard regularization without self-consistency; and for individual adsorbents using the
model of slitshaped pores for PS1 and the model of gaps between spherical particles
(cubic lattice) for fumed silica.
A-300/Si-60/Envicarb
Sph/Cyl/Slit
Slit
Individual
A-300(Sph)
Si-60(Cyl)
Envicarb(Slit)
IPSDV (a.u.)
0.02
0.01
0.00
1
10
100
Pore Radius (nm)
Fig. 2. Incremental PSDs for mechanical mixture of fumed silica A-300, silica gel Si-60
and carbon black Envicarb (1 : 1 : 1) calculated using a complex model of pores with
contribution of gaps between spherical particles (A-300), cylindrical pores (Si-60) and
slitshaped pores (Envicarb); and for individual adsorbents using the model of: gaps
between spherical particles (cubic lattice) for fumed silica; cylindrical pores for Si-60,
and slitshaped pores for Envicarb. Additionally, the model of slitshaped pores was used
for the mixture.
252
V. M. Gun’ko and R. Leboda
Additionally, the accessibility of the surfaces of different components can differ
from their percentage. The SCR method can improve the PSD for complex
systems because it gives a better fitting of the isotherm. In this approach,
contributions of pores of different shapes are varied for the best correspondence
of the theoretical isotherm to experimental one. The developed procedure was
used [5] to describe such adsorption characteristics as adsorption energy
distributions [11, 39-42]. Consideration for the difference in the nature of
components of complex adsorbents is very important for an accurate description
of the energetic characteristics of adsorbents.
Notice that the developed MNDC procedure brings the PSD close to that
computed one using the DFT method (Figs 3 and 4).
0.05
WVA
DFT
MNDC
IPSDV (a.u.)
0.04
0.03
0.02
0.01
0.00
0
5
10
15
20
Pore Half-width (nm)
Fig. 3. PSDs for activated carbon WVA [17] calculated using the DFT and MNDC
methods.
IPSDV (a.u.)
0.03
Carboxen 569
MNDC
DFT
0.02
0.01
0.00
0
20
40
60
80
100
120
Pore Half-width (nm)
Fig. 4. PSDs for activated carbon Carboxen 569 (Supelco) calculated using the DFT and
MNDC methods.
Pore size distributions of complex systems
253
The deviation of the pore shape from the model is small ∆wtotal = 0.043 for the
mixed model with slitshaped pores and gaps between spherical particles applied
to the mixture with A-300/PS1 (Fig. 1) and ∆wslit = 0.507 in the case of the use
of slitlike pore model for this complex adsorbent.
4. CONCLUSIONS
The developed procedure with SCR gives more reliable pore size
distributions for complex adsorbents than standard methods. The later are based
on the integral isotherm equation including certain adsorption potential for a
given pore model developed for individual adsorbents and, therefore,
inappropriate for hybrid adsorbents including texturally and chemically different
components.
Acknowledgments. This research was supported by NATO (grant
PST.CLG.979895). R. Leboda is grateful to the Foundation for Polish Science
for financial support.
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Pore size distributions of complex systems
255
CURRICULA VITAE
Vladimir Gun’ko was born in Ukraine in 1951. He received
his M.Sc. in Theoretical Physics from Dnipropetrovsk State
University in 1973. Gun’ ko received his Ph.D. degree in
Chemistry from the Institute of Physicoorganic Chemistry and
Coil Chemistry (Kiev) in 1983 and Sc.D. in Physics and
Chemistry (1995) from the Institute of Surface Chemistry,
National Academy of Sciences of Ukraine in Kiev.
Professional experience: 1973-1976 – Engineer at the
Dnipropetrovsk State University; 1976-1978 – Senior Engineer
and 1978-1985 – Junior Researcher at the Institute of
Physicoorganic Chemistry and Coil Chemistry (Kiev); 1985-1991 – Senior Researcher,
1991-1996 – Head of Lab, and from 1996 – Leading Researcher at the Institute of Surface
Chemistry (Kiev). He is a member of Chemical Society of Ukraine. He was a visiting
professor at the University of Brighton (UK) (2001 and 2003), Maria Curie-Skłodowska
University (2004), and Technical University of Athens (1998). He visited the MCSU
dozen times to carry out joint investigations and published more than hundred joint
papers with colleagues from Poland. Gun’ ko is a member of the Editorial Board of the
journal “Theoretical and Experimental Chemistry” (Kiev).
Specialization: Theoretical Chemistry, Quantum Chemistry, Physical Chemistry, Colloid
Chemistry, Applied Mathematics, and programming.
Current research interest: adsorption, interfacial phenomena.
Number of papers in referred journals: about 230.
Number of communications to scientific meetings: about 65.
Roman Leboda was born in Poland in 1943. Graduated from
Maria Curie-Skłodowska University (MCSU) in Lublin (1967).
He obtained the Ph.D. and Sc.D. degrees in 1974 and 1981,
respectively, from MCSU. He received the professor title in
1989 at the MCSU. He is a member of Polish Chemical Society
and International Adsorption Society, Visiting Professor in the
Institute of Inorganic and Analytical Chemistry, Gutenberg
University (Mainz, 1982-1983). He was President of the Lublin
Division of Polish Chemical Society in 1989-1993. He
organized seven Polish-Ukrainian Symposia on Theoretical and
Experimental Studies of Interfacial Phenomena and Their Technological Applications.
Specialization: Physical Chemistry, Chromatography, Physical Chemistry of Surface,
Environmental Chemistry, Adsorption.
Current research interests: synthesis and modification of carbon and carbon-mineral
adsorbents, textural and adsorption characterization and applications of these materials;
analysis of trace amounts of substances in air, water and soil.
Number of books: 2.
Number of papers in referred journals: about 300 and 30 patents.
Number of communications to scientific meetings: about 170.